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Miguel is planning to invest $5,000. He plans to withdraw his investment and earnings in approximately 10 years. With Option #1, his money will grow according to the function p(x)=5000⋅1.06x. With Option #2, his money will grow accordingly to the function q(x)=500x+5,000

. He creates sequences for both functions.
(2 points)

Which option has the greater rate of change between years 7 and 8? Option #
Which option should he avoid so he can have more money? Option #

1 answer

To analyze Miguel's investment options, we will look at the growth of each option according to the functions provided and calculate the rate of change between years 7 and 8.

Option #1: \( p(x) = 5000 \cdot 1.06^x \)

  1. Calculate \( p(7) \): \[ p(7) = 5000 \cdot 1.06^7 \approx 5000 \cdot 1.50363 \approx 7518.15 \]

  2. Calculate \( p(8) \): \[ p(8) = 5000 \cdot 1.06^8 \approx 5000 \cdot 1.59385 \approx 7969.24 \]

  3. Calculate the rate of change between years 7 and 8: \[ \text{Rate of Change (Option #1)} = p(8) - p(7) = 7969.24 - 7518.15 \approx 451.09 \]

Option #2: \( q(x) = 500x + 5000 \)

  1. Calculate \( q(7) \): \[ q(7) = 500 \cdot 7 + 5000 = 3500 + 5000 = 8500 \]

  2. Calculate \( q(8) \): \[ q(8) = 500 \cdot 8 + 5000 = 4000 + 5000 = 9000 \]

  3. Calculate the rate of change between years 7 and 8: \[ \text{Rate of Change (Option #2)} = q(8) - q(7) = 9000 - 8500 = 500 \]

Summary:

  • Rate of Change for Option #1 between years 7 and 8: 451.09
  • Rate of Change for Option #2 between years 7 and 8: 500

Conclusions:

  1. Which option has the greater rate of change?

    • Option #2 has the greater rate of change (500 > 451.09).

  2. Which option should he avoid to have more money?

    • Option #1 should be avoided because it has a lower rate of change, and hence, it will generally yield less money over time compared to Option #2.